Упр.1109 ГДЗ Алимов 10-11 класс (Алгебра)
- Найти значение выражения, предварительно его упростив:
1) $$\mathrm{C}_{4}^{1}+\mathrm{C}_{4}^{2}+\mathrm{C}_{4}^{3}+\mathrm{C}_{4}^{4}$$;
2) $$\mathrm{C}_{5}^{1}+\mathrm{C}_{5}^{2}+\mathrm{C}_{5}^{3}+\mathrm{C}_{5}^{4}$$;
3) $$\mathrm{C}_{7}^{0}+\mathrm{C}_{7}^{7}+\mathrm{C}_{7}^{1}+\mathrm{C}_{7}^{6}+\mathrm{C}_{7}^{2}+\mathrm{C}_{7}^{5}+\mathrm{C}_{7}^{3}+\mathrm{C}_{7}^{4}$$;
4) $$\mathrm{C}_{6}^{0}+\mathrm{C}_{6}^{6}+\mathrm{C}_{6}^{1}+\mathrm{C}_{6}^{5}+\mathrm{C}_{6}^{2}+\mathrm{C}_{6}^{4}+\mathrm{C}_{6}^{3}$$;
5) $$\mathrm{C}_{12}^{2}+\mathrm{C}_{12}^{3}+\mathrm{C}_{13}^{4}+\mathrm{C}_{14}^{5}$$;
6) $$\mathrm{C}_{9}^{3}+\mathrm{C}_{9}^{4}+\mathrm{C}_{10}^{5}+\mathrm{C}_{11}^{6}$$;
7) $$\mathrm{C}_{21}^{7}-\mathrm{C}_{20}^{7}$$;
8) $$\mathrm{C}_{25}^{9}-\mathrm{C}_{24}^{9}$$;
9) $$\mathrm{C}_{15}^{10}-\mathrm{C}_{14}^{9}$$;
10) $$\mathrm{C}_{13}^{8}-\mathrm{C}_{12}^{7}$$.
Используем свойство биномиальных коэффициентов:
$$C_n^k + C_n^{k+1} = C_{n+1}^{k+1}$$
и симметрию:
$$C_n^k = C_n^{n-k}.$$
$$C_4^1 + C_4^2 + C_4^3 + C_4^4 = \left(C_4^0 + C_4^1 + C_4^2 + C_4^3 + C_4^4\right) — C_4^0 = 2^4 — 1 = 15.$$
$$C_5^1 + C_5^2 + C_5^3 + C_5^4 = \left(C_5^0 + C_5^1 + C_5^2 + C_5^3 + C_5^4 + C_5^5\right) — C_5^0 — C_5^5 = 2^5 — 1 — 1 = 30.$$
$$C_7^0 + C_7^7 + C_7^1 + C_7^6 + C_7^2 + C_7^5 + C_7^3 + C_7^4 = \sum_{k=0}^{7} C_7^k = 2^7 = 128.$$
$$C_6^0 + C_6^6 + C_6^1 + C_6^5 + C_6^2 + C_6^4 + C_6^3 = \sum_{k=0}^{6} C_6^k = 2^6 = 64.$$
$$C_{12}^2 + C_{12}^3 + C_{12}^4 + C_{12}^5 = C_{13}^3 + C_{13}^4 + C_{13}^5 = C_{14}^4 + C_{14}^5 = C_{15}^5.$$
$$C_{15}^5 = \frac{15!}{5!\,10!} = \frac{15\cdot 14\cdot 13\cdot 12\cdot 11}{5\cdot 4\cdot 3\cdot 2} = 3003.$$
$$C_9^3 + C_9^4 + C_9^5 + C_9^6 = C_{10}^4 + C_{10}^5 + C_{10}^6 = C_{11}^5 + C_{11}^6 = C_{12}^6.$$
$$C_{12}^6 = \frac{12!}{6!\,6!} = \frac{12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7}{6\cdot 5\cdot 4\cdot 3\cdot 2} = 924.$$
$$C_{21}^7 — C_{20}^7 = C_{20}^6.$$
$$C_{20}^6 = \frac{20!}{6!\,14!} = \frac{20\cdot 19\cdot 18\cdot 17\cdot 16\cdot 15}{6\cdot 5\cdot 4\cdot 3\cdot 2} = 38760.$$
$$C_{25}^9 — C_{24}^9 = C_{24}^8.$$
$$C_{24}^8 = \frac{24!}{8!\,16!} = \frac{24\cdot 23\cdot 22\cdot 21\cdot 20\cdot 19\cdot 18\cdot 17}{8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2} = 735471.$$
$$C_{15}^{10} — C_{14}^9 = C_{14}^{10}.$$
$$C_{14}^{10} = C_{14}^4 = \frac{14!}{4!\,10!} = \frac{14\cdot 13\cdot 12\cdot 11}{4\cdot 3\cdot 2} = 1001.$$
$$C_{13}^8 — C_{12}^7 = C_{12}^8.$$
$$C_{12}^8 = C_{12}^4 = \frac{12!}{4!\,8!} = \frac{12\cdot 11\cdot 10\cdot 9}{4\cdot 3\cdot 2} = 495.$$
Ответ: 1) $$15$$; 2) $$30$$; 3) $$128$$; 4) $$64$$; 5) $$3003$$; 6) $$924$$; 7) $$38760$$; 8) $$735471$$; 9) $$1001$$; 10) $$495$$.
